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Theorem rspesbca 2988
 Description: Existence form of rspsbca 2987. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
rspesbca ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rspesbca
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2907 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
21rspcev 2784 . 2 ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑦𝐵 [𝑦 / 𝑥]𝜑)
3 cbvrexsv 2664 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑦 / 𝑥]𝜑)
42, 3sylibr 133 1 ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480  [wsb 1735  ∃wrex 2415  [wsbc 2904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905 This theorem is referenced by:  spesbc  2989
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